offer that is less. So, if the buyer tenders an offer of five million dollars, then the dotcom owners will accept if their value is between zero and five million. The buyer, being
strategic, then realizes that this implies the value of the company to her

20, the payoff 90 in Figure 6 should be changed to 200. The units in which payoffs
are measured are arbitrary. Like degrees on a temperature scale, they can be multiplied
by a positive number and shifted by adding a constant, without altering the underlyi

lutionary game. Unlike Figure 5, it is a non-symmetric interaction between a vendor who
chooses Dont Inspect and Inspect for certain fractions of a large number of interactions.
Player IIs actions comply and cheat are each chosen by a certain fraction of

II
H
(2, 2)
HH
H
dont HHH
H (0, 1)
buy
High
I
buy
*
@
@
@
buy
Low @
@
@
HH
II
H
(3, 0)
H
HH
j
dont HHH
H (1, 1)
buy
Figure 7. Quality choice game where player I commits to High or Low quality, and
player II can react accordingly. The arrows indicate the

gies of the players. In the game tree, any strategy combination results into an outcome
of the game, which can be determined by tracing out the path of play arising from the
players adopting the strategy combination. The payoffs to the players are then en

game. The analysis of dynamic strategic interaction was pioneered by Selten, for which
he earned a share of the 1994 Nobel prize.
First-mover advantage
A practical application of game-theoretic analysis may be to reveal the potential effects
of changing t

correct comparison is to consider commitment to a randomized choice, like to a certain
inspection probability. In Figure 6, already the commitment to the pure strategy Inspect
gives a better payoff to player I than the original mixed equilibrium since pla

high, medium, low, or none at all, denoted by H, M, L, N for firm I and h, m, l, n for
firm II. The market price of the memory chips decreases with increasing total quantity
produced by both companies. In particular, if both choose a high quantity of prod

The payoffs in Figure 9 are derived from the following simple model due to Cournot.
The high, medium, low, and zero production numbers are 6, 4, 3, and 0 million memory
chips, respectively. The profit per chip is 12 Q dollars, where Q is the total quantit

developed the basis for a strong competing product. For brevity, when the large company
has the ability to produce a strong competing product, the company will be referred to as
having a strong position, as opposed to a weak one.
The large company, after

Strategies in extensive games
In an extensive game with perfect information, backward induction usually prescribes
unique choices at the players decision nodes. The only exception is if a player is indifferent between two or more moves at a node. Then, an

(20, 4)
stay
in
0.5
chance
(Announce) HH
H
H
sell HHH
I
H
out
II
(12, 4)
( 4, 20)
stay
in
@
@
HH
Announce
H
HH
0.5@
sell
HH
I
H (12, 4)
@
out
@
H
H
H
HH
Cede HH
H
(0, 16)
Figure 10. Extensive game with imperfect information between player I, a

indifferent, receiving an overall expected payoff of 9 in each case. This can also be seen
from the extensive game in Figure 10: when in a weak position, player I is indifferent
between the moves Announce and Cede where the expected payoff is 0 in each ca

A similar use of randomization is known in the theory of algorithms as Raos theorem,
and describes the power of randomized algorithms. An example is the well-known quicksort algorithm, which has one of the best observed running times of sorting algorithms

While second-price sealed-bid auctions like the one described above are not very common, they provide insight into a Nash equilibrium of the English auction. There is a
strategy in the English auction which is analogous to the weakly dominant strategy in

the potential buyers present, an auctioneer raises the price for the object as long as two
or more bidders are willing to pay that price. The auction stops when there is only one
bidder left, who gets the object at the price at which the last remaining op

information set, which has two nodes according to the different histories of play, which
player II cannot distinguish.
Because player II is not informed about its position in the game, backward induction
can no longer be applied. It would be better to sel

different results, and, ideally, each bidder would like to have access to all the surveys in
formulating its bid. Since the information is proprietary, that is not possible.
Strategic thinking, then, requires the bidders to take into account the additiona

them (that is, play them randomly) without losing payoff. The only case where, in turn,
the original mixed strategy of player I is a best response is if player I is indifferent. According to the payoffs in Figure 6, this requires player II to choose compl

Mixed equilibrium
What should the players do in the game of Figure 6? One possibility is that they prepare
for the worst, that is, choose a max-min strategy. As explained before, a max-min strategy
maximizes the players worst payoff against all possible c

@ II
I@
Dont
inspect
comply cheat
10
0
0
10
0
90
Inspect
1
6
Figure 6. Inspection game between a software vendor (player I) and consumer (player II).
is that the vendor chooses Dont inspect and the consumer chooses to comply. Without
inspection, the cons

3 Dominance
Since all players are assumed to be rational, they make choices which result in the outcome they prefer most, given what their opponents do. In the extreme case, a player may
have two strategies A and B so that, given any combination of strate

Game Theory
Theodore L. Turocy
Texas A&M University
Bernhard von Stengel
London School of Economics
CDAM Research Report LSE-CDAM-2001-09
October 8, 2001
Contents
1
What is game theory?
4
2 Definitions of games
6
3
Dominance
8
4
Nash equilibrium
12
5
Mixe

Glossary
Backward induction
Backward induction is a technique to solve a game of perfect information. It first considers the moves that are the last in the game, and determines the best move for the player
in each case. Then, taking these as given future

Mixed strategy
A mixed strategy is an active randomization, with given probabilities, that determines the
players decision. As a special case, a mixed strategy can be the deterministic choice of
one of the given pure strategies.
Nash equilibrium
A Nash eq

Strategy
In a game in strategic form, a strategy is one of the given possible actions of a player. In
an extensive game, a strategy is a complete plan of choices, one for each decision point
of the player.
Zero-sum game
A game is said to be zero-sum if fo

2 Definitions of games
The object of study in game theory is the game, which is a formal model of an interactive
situation. It typically involves several players; a game with only one player is usually
called a decision problem. The formal definition lays

players making choices out of their own interest. Cooperation can, and often does, arise
in noncooperative models of games, when players find it in their own best interests.
Branches of game theory also differ in their assumptions. A central assumption in

in economic theory. Additionally, it has found applications in sociology and psychology,
and established links with evolution and biology. Game theory received special attention
in 1994 with the awarding of the Nobel prize in economics to Nash, John Harsa

No rational player will choose a dominated strategy since the player will always be
better off when changing to the strategy that dominates it. The unique outcome in this
game, as recommended to utility-maximizing players, is therefore (D, d) with payoffs